{
  "abstract": "We construct and study the Nash manifold of four-point configurations in the real plane modulo the action of a Nash subgroup of the group of similarity transformations. We do so by using the finer notion of a quadrangle in place of that of a four-point configuration, since this retains limiting line data in degenerations. We prove that the space ${\\mathcal Q}$ of quadrangles is an 8-dimensional Nash manifold and that, for every Nash subgroup $G$ of the similarity group, the orbit space ${\\mathcal Q}/G$ is a Nash manifold. We define a geometric invariant, called aspect, with values in $[-1,1]$, and prove that, over each of the intervals $(-1,0)$ and $(0,1)$, the corresponding part of ${\\mathcal Q}/G$ is Nash diffeomorphic to the product of the interval with a fixed fiber. Thus the nonexceptional part of the moduli problem reduces to the analysis of two model fibers, each having a natural geometric interpretation in terms of the quadrangles themselves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02087",
  "author_total": 2,
  "authors": [
    {
      "name": "Bruce Olberding"
    },
    {
      "name": "Elaine A. Walker"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ez3s9pjfjfnxf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02087",
  "fulltext_url": null,
  "id": "ez3s9pjfjfnxf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02087",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Mathematics and Applications"
    ]
  ],
  "openalex": "W7216425891",
  "pdf_url": "https://arxiv.org/pdf/2610.02087",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ez3s9pjfjfnxf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.02087",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02087"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Nash manifold of four-point configurations modulo similarity subgroups",
  "title_text": "The Nash manifold of four-point configurations modulo similarity subgroups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/ez3s9pjfjfnxf",
  "venue": "arXiv",
  "versions": []
}